We note that Eq. (REF ) does not automatically impose SU(2), translational, and point-group symmetries, which are challenging to implement in sizeable fermionic systems. Therefore, this bound is usually not strict for doped fermionic systems and for 2-producible SPFM states we find it to be \(\sim 1.5\) , i.e. much higher than the value obtained for the spin modes in Fig. REF . Crucially, since the QFI bounds represent a sufficient but not necessary condition for the presence of entanglement, the overall scale of the QFI cannot exclude the presence of other highly-entangled states and the entanglement depth should be estimated via the highest available \(k\)  witnessed by different QFIs. A more accurate and basis-independent definition of entanglement in fermionic systems requires considering indistinguishable particles subject to anticommutation relations, and the superselection rule [1]}, [2]}. Entanglement measures satisfying these conditions, such as the the Slater rank [3]}, [4]}, usually require high-order correlation functions beyond the ones accessible with spectra in the form of Eq. (REF ). Therefore, distilling entanglement in the fermionic basis from spectroscopic measurements remains an open and interesting question.
